Stationary Ergodic Data Research Articles

Some asymptotic results of a kNN conditional mode estimator for functional stationary ergodic data

. This article investigates the conditional mode estimation of a univariate response variable given a functional random covariate (i.e., valued in some infinite-dimensional space) whenever a stationary ergodic data are considered. We construct a new estimator of the conditional mode which is constructed by a kNN approach. Under less restrictive conditions, we show the strong consistency of the proposed estimator. To assess the efficiency of the developed estimator, empirical analysis as well as real data analyses are performed.

Uniform limit theorems for a class of conditional [formula omitted]-estimators when covariates are functions

This paper considers nonparametric estimation of a parameter, which is a zero of a certain estimating equation, indexed by a class of functions and depending on an infinite-dimensional covariate. This problem is known in the literature as Z-estimation, where a rich theory, in the case of the finite-dimensional covariates, exists. However, in certain applications, covariates may belong to an infinite-dimensional space, which makes the estimation as well as the study of the asymptotic properties challenging topics. This paper aims to bring several contributions to the existing functional data analysis literature. First, we introduce nonparametric conditional Z-estimators whenever functional stationary ergodic data are observed under right-censorship model. A sharp uniform-in-bandwidth limit law for the proposed estimator is presented. Such result allows the bandwidth to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. Moreover, a uniform consistency rate, over Vapnik–Chervonenkis classes of functions, is provided for the process of conditional Z-estimators. Finally, a uniform central limit theorem for the process indexed by functions is established. The last result is of particular interest since it is obtained for dependent functional data. The uniform limit theorems, discussed in this paper, are key tools for many further developments in functional data analysis involving empirical process techniques. These results are proved under some standard structural conditions on the Vapnik–Chervonenkis classes of functions and some mild conditions on the model.

Nonparametric M-estimation for Functional Stationary Ergodic Data

This paper considers a nonparametric M-estimator of a regression function for functional stationary ergodic data. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. Under some mild conditions, the weak consistency and the asymptotic normality of the M-estimator are established. Furthermore, a simulated example is provided to examine the finite sample performance of the M-estimator.

The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions

We consider the double functional regression model Y=r(X)+ε, where the response variable Y is Hilbert space-valued and the covariate X takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields.We study a kernel estimator of the Nadaraya–Watson type for the operator r and derive its limiting law which is a Gaussian operator on the Hilbert space. Moreover, we investigate both a naive and a wild bootstrap procedure in the double functional setting and demonstrate their asymptotic validity. This is quite useful as building confidence sets based on an asymptotic Gaussian distribution is often difficult.

The kernel regression estimation for randomly censored functional stationary ergodic data

ABSTRACTIn this paper, we investigate the asymptotic properties of the kernel estimator for non parametric regression operator when the functional stationary ergodic data with randomly censorship are considered. More precisely, we introduce the kernel-type estimator of the non parametric regression operator with the responses randomly censored and obtain the almost surely convergence with rate as well as the asymptotic normality of the estimator. As an application, the asymptotic (1 − ζ) confidence interval of the regression operator is also presented (0 < ζ < 1). Finally, the simulation study is carried out to show the finite-sample performances of the estimator.

Nonparametric [formula omitted]-estimation for right censored regression model with stationary ergodic data

The present paper deals with a nonparametric M-estimation for right censored regression model with stationary ergodic data. Defined as an implicit function, a kernel-type estimator of a family of robust regression is considered when the covariate takes its values in Rd (d≥1) and the data are sampled from a stationary ergodic process. The strong consistency (with rate) and the asymptotic distribution of the estimator are established under mild assumptions. Moreover, a usable confidence interval is provided which does not depend on any unknown quantity. Our results hold without any mixing condition and do not require the existence of marginal densities. A comparison study based on simulated data is also provided.

Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ⊝φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θφ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θφ(x) - Θφ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.

Rate of uniform consistency for a class of mode regression on functional stationary ergodic data

The aim of this paper is to study the asymptotic properties of a class of kernel conditional mode estimates whenever functional stationary ergodic data are considered. To be more precise on the matter, in the ergodic data setting, we consider a random elements (X, Z) taking values in some semi-metric abstract space $$E\times F$$ . For a real function $$\varphi $$ defined on the space F and $$x\in E$$ , we consider the conditional mode of the real random variable $$\varphi (Z)$$ given the event “ $$X=x$$ ”. While estimating the conditional mode function, say $$\theta _\varphi (x)$$ , using the well-known kernel estimator, we establish the strong consistency with rate of this estimate uniformly over Vapnik–Chervonenkis classes of functions $$\varphi $$ . Notice that the ergodic setting offers a more general framework than the usual mixing structure. Two applications to energy data are provided to illustrate some examples of the proposed approach in time series forecasting framework. The first one consists in forecasting the daily peak of electricity demand in France (measured in Giga-Watt). Whereas the second one deals with the short-term forecasting of the electrical energy (measured in Giga-Watt per Hour) that may be consumed over some time intervals that cover the peak demand.

Conditional mode estimation for functional stationary ergodic data with responses missing at random

In this paper, we investigate the asymptotic properties of a non-parametric conditional mode estimation given a functional explanatory variable, when functional stationary ergodic data and missing at random responses are observed. First of all, we establish asymptotic properties for a conditional density estimator from which we derive almost sure convergence (with rate) and asymptotic normality of a conditional mode estimator. This new estimate take into account missing data, and a simulation study is performed to illustrate how this fact allows to get higher predictive performances than those obtained with standard estimates.

Uniform in bandwidth rate of convergence of the conditional mode estimate on functional stationary ergodic data

The aim of this paper is to establish the uniform consistency with rate over a bandwidth interval of the kernel conditional mode estimate whenever functional stationary ergodic data are considered. This kind of result is immediately applicable to proving uniform consistency of kernel-type estimators when the bandwidth h is a function of the data or the location x. Notice that our uniform in bandwidth results are the first ones to be established in this setting. Moreover, the ergodic setting offers a more general framework in regards to the practice than the usual mixing structure.

Nonparametric regression estimation for functional stationary ergodic data with missing at random

In this paper, we investigate the asymptotic properties of the estimator for the regression function operator whenever the functional stationary ergodic data with missing at random (MAR) are considered. Concretely, we construct the kernel type estimator of the regression operator for functional stationary ergodic data with the responses MAR, and some asymptotic properties such as the convergence rate in probability as well as the asymptotic normality of the estimator are obtained under some mild conditions respectively. As an application, the asymptotic (1−ζ) confidence interval of the regression operator is also presented for 0<ζ<1. Finally, a simulation study is carried out to compare the finite sample performance based on mean square error between the classical functional regression in complete case and the functional regression with MAR.

Randomly censored quantile regression estimation using functional stationary ergodic data

This paper investigates the conditional quantile estimation of a randomly censored scalar response variable given a functional random covariate (i.e. valued in some infinite-dimensional space) whenever a stationary ergodic data are considered. A kernel-type estimator of the conditional quantile function is introduced. Then, a strong consistency rate as well as the asymptotic distribution of the estimator are established under mild assumptions. A simulation study is considered to show the performance of the proposed estimator. An application to the electricity peak demand prediction using censored smart meter data is also provided.

Vector-on-function quantile regression for stationary ergodic processes

This paper deals with quantile regression of a vector response (Y∈Rq, q≥2) on a functional covariate X that takes values in an infinite dimensional space. The main purpose is to introduce a kernel-type estimator of the conditional geometric quantiles whenever functional strictly stationary ergodic data are considered. We established the strong consistency with rate of the proposed estimator as well as the asymptotic distribution that leads to build a confidence region for the multivariate quantile regression. We gave also an application to joint horizon time series forecasting.

Empirical Likelihood Inference for Nonparametric Regression Functions with Functional Stationary Ergodic Data

In this article, we employ the empirical likelihood method to construct confidence intervals for nonparametric regression functions with functional stationary ergodic data. Under some wild conditions, it is proved that the empirical likelihood ratio statistic is asymptotically chi-squared distributed.

Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties

The aim of this paper is to study asymptotic properties of the kernel regression estimate whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. While estimating the regression function using the well-known Nadaraya–Watson estimator, we establish the consistency in probability, with a rate, as well as the asymptotic normality which induces a confidence interval for the regression function usable in practice since it does not depend on any unknown quantity. We also give the explicit form of the conditional bias term. Note that the ergodic framework is more convenient in practice since it does not need the verification of any condition as in the mixing case for example.

Rates of strong consistencies of the regression function estimator for functional stationary ergodic data

The aim of this paper is to study both the pointwise and uniform consistencies of the kernel regression estimate and to derive also rates of convergence whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric separable abstract space. While estimating the regression function using the well-known Nadaraya–Watson estimator, we establish the strong pointwise and uniform consistencies with rates. Depending on the Vapnik–Chervonenkis size of the class over which uniformity is considered, the pointwise rate of convergence may be reached in the uniform case. Notice, finally, that the ergodic data framework extends the dependence setting to cases that are not covered by the usual mixing structures.

WINDOW AND OVERLAP PROCESSING EFFECTS ON POWER ESTIMATES FROM SPECTRA

Fast Fourier transform (FFT) spectral processing is based on the assumption of stationary ergodic data. In engineering practice, the assumption is often violated and non-stationary data processed. Data windows are commonly used to reduce leakage by decreasing the signal amplitudes near the boundaries of the discrete samples. With certain combinations of non-stationary signals and windows, the temporal weighting may attenuate important signal characteristics to adversely affect any subsequent processing. In other words, the window artificially reduces a significant section of the time signal. Consequently, spectra and overall power estimated from the affected samples are unreliable. FFT processing can be particularly problematic when the signal consists of randomly occurring transients superimposed on a more continuous signal. Overlap processing is commonly used in this situation to improve the estimates. However, the results again depend on the temporal character of the signal in relation to the window weighting. A worst-case scenario, a short-duration half sine pulse, is used to illustrate the relationship between overlap percentage and resulting power estimates. The power estimates are shown to depend on the temporal behaviour of the square of overlapped window segments. An analysis shows that power estimates may be obtained to within 0.27 dB for the following windows and overlap combinations: rectangular (0% overlap), Hanning (62.5% overlap), Hamming (60.35% overlap) and flat-top (82.25% overlap).

Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression

Theorems concerning the entropy of a stationary ergodic information source are derived and used to obtain insight into the workings of certain data-compression coding schemes, in particular the Lempel-Siv data compression algorithm. >